JOURNAL OF MATERIALS SCIENCE 31 (1996) 3109-3114

Determination of the hardness and elasticmodulus from continuous Vickersindentation testing

J. GUBICZA, A. JUHÁSZ, P. TASNÁDI, P. ARATÖ*, G. VÖRÖSDepartment of General Physics, Eötvös University, Budapest, H-l088 Budapest,Múzeum krt. 6-8 Hungary*Research Institute for Technical Physics, P.D.B. 76, H-1325, Budapest, Hungary

Continuous Vickers (Hv) indentation tests were performed on different materials (ioncrystals, metals, ceramics, silica glass and plastic). Load-indentation depth curves weretaken during the loading as weil as during the unloading period by a computer controlledhydraulic mechanical testing machine (MTS 810). The indentation work measured both theloading and the unloading periods, and these were used for the evaluation of parameterscharacterizing the materials. It was found empirically that there were linear connectionsbetween the maximum load to the power 312 and the indentation work. These connectionswere used to relate the conventional hardness number, Hv, and Young's modulus, E, withthe work performed during loading and unloading. This work can be determined with greataccuracy from the measurements. The values of the Young's modulus and the Vickershardness determined this way agree weil with those obtained by conventional methods. Onthe basis of continuous indentation tests, materials can be easily classified into theisomechanical groups introduced by Ashby. For this classification the Hvl E ratio is generallyused. As a substitute for Hvl E another parameter is recommended which can be determinedeasily from a single measurement.

;,

f

1. IntroductionVarious types of continuous indentation tests havecome into general use for the determination ofmechanical properties of materials. The indentationmethod is preferred because relatively small amountsof testing material are needed and there are no strictrequirements for the share of the samples, moreoverthe measurements can be perforrned without the de-struction of the samples. For these investigationsa wide variety of testing devices were developed withindenters of various forms working in a scale from thenanoidentation to macrohardness region. The com-mon feature of these tests is that the applied load isregistered as a function of indentation depth duringboth the loading and unloading period [1-6].

A schematic load-penetration depth curve is shownin Fig. 1.The curves taken during loading and unIoad-ing can be described by polynoms [1, 3-6J or powerlaw functions [2]. The parameters of these functionsdepend intricately on both the elastic and the plasticproperties of the samples investigated because mater-ials are generally deformed both elastically and plasti-Gally by a sharp indentor. Therefore, to find aut theconventional elastic and plastic parameters of a ma-terial from the continuous indentation tests is nota simple problem. However, there are various theoret-icaIly more or less well supported methods for theevaluation of the load-penetration depth functionsobtained experimentally.

0022-2461 eg 1996 Chapman & Hall

Fröhlich et al. [lJ studying the connection betweenthe hardness number and the parameters of the inden-tation curve described the loading period by a quad-ratic polynom. He proposed that the coefficient of thequadratic term can be regarded as a new hardnessnumber characterizing the resistance of the bulk ma-terial against elastic-plastic deformation. Themain advantage of the use of this parameter is that itis independent of the magnitude ofthe load and conse-quently from the diagonal of the Vickers pattern. Asis well known, the conventional hardness number isby definition the maximum load divided by the con-tact area of the indenter and the sampIe measuredat maximum load [7-10]. Direct calculation ofthe conventional hardness number from a continuousindentation test is difficult because generally thereis no simple way for the determination of the contactarea. Oliver and Ph arr [2J elaborated on an iterativeprocedure for the determination of the projectedarea of the contact surface. It needed the maximumpenetration depth and the slope of the unloadingcurve at its starting point as initial parameters. Theuncertainty of both parameters is relatively high. Forthe first quantity it is the consequence of the uncer-tainty of the measurement of the load, for the secondone it is due to the fact that the calculated value ofthe slope of the unloading curve depends stronglyon the form of the function fitted to the unloadingcurve.

3109

Methods available in the literature for the calcu-lation of Young's modulus from indentation tests arebased in most cases on the theoretical solution of theBoussinesq problem given by Sneddon [llJ who de-termined the indentation depth within an elastichalf space for various indenters. Ph arr et al. [12J,using Sneddon's results, have calculated the reducedYoung'smodulus, En [(Er= Ell - V2) where E is theYoung's modulus and v is the Poisson's ratio J fora broad variety of materials. Their input data were inthis case also the slope ofthe initial part of the unIoad-ing curve and the contact area of the indenter and thesampIe at the maximum load.

Sakai [4J applied a Maxwell type elastic-plasticmodel for the description of the indentation process.He defined a so-called "true hardness" for the charac-terization of the plastic properties of the materials anddetermined it from the energy dissipated duringa loading-unloading period. Sakai also determinedthe reduced Young's modulus from the unloadingcurve; however, he did not take into account that thiscurve depended not only on the elastic but also on theplastic properties of the materials.

In the present paper a detailed analysis of the con-tinuous indentation curves is given and a new methodbased on energetic argumentation is propo sed forthe determination of the hardness number and theYoung's modulus.

2. Experimental procedureThe mechanical properties of the materials were inves-tigated by a special kind of continuous Vickers hard-ness test. The surface of the samples was mechanicallypolished before measurement. During the test aVickers pyramid was pressed into the surface of thesampIe by a computer controlled hydraulic mechan-ical testing machine (MTS 810). The measurementswere carried aut in the macrohardness region(Pmax ~ 100 N). During the loading period the Vickerspyramid penetrated into the surface of the sampIe atconstant velocity and the same velocity was applied inthe unloading period when the pyramid muved back-wards. In the course of the test the load was registeredas a function of the penetration depth. Measurementswere perforrned on various materials: metals (99.99%pure Al and Cu), soda lime silica glass, aIkali halogen-ide crystals, polypropylene, Si3N4 ceramics of a com-position 90 wt % Si3Nc4 wt % A12O3-6wt % Y203sintered to different densities and a series of alumina-hydroxyapatite ceramic composites (AI2OrHAP)with alumina contents up to 60 vol %, and a tetra-gonal zirconia polycrystal ceramic sampIe containing10 mol % Ce02 (Ce- TZP ceramic).

3. Results and discussion3.1. Features of the indentation curvesIn the loading period the load-penetration depthfunction can be described by a quadratic polynom(Fig. 1)

P = c2h + c3h2

3110

For the unloading period the load also satisfies aquadratic equation

P = ci(h - ho) + c~(h - hof (2)

where P is the load; h is the penetration depth; ho is theresidual indentation depth after removing the punch;and C2, C3, ci, c~ are fitting parameters. The totalindentation work is the integral of the load with re-spect to the indentation depth and it equals to the areaunder the load-penetration depth curve correspond-ing to the loading period. During unloading a portionof this work can be regained, it is equal to the areaunder the load-indentation depth curve for this latterperiod.

The difference between these two calculations forwork gives the energy dissipated during the load-ing-unloading period (Fig. 1). The work perforrnedduring loading, Wt, and unloading, W., can be cal-culated by integration of Equations 1 and 2, respec-tively

W = C2h2 + C3h3t 2 m 3 m

(3)

* *C2 2 C3 3

We = 2(hm - ho) + 3(hm - ho) (4)

where hm is the maximum indentation depth. Thedifference between the two quantities equals the workdissipated, i.e.

Wd - Wt - We (5)

If the load-penetration depth functions were quad-ratic (C2= ci = O), ali three calculations for workshould be proportional to p;j2. In spite of the fact thatthe experimental results show that these functions arenot quadratic for the materials investigated, the linearrelationship between the works and p;j2 is still a goodapproximation. This fact is illustrated in Fig. 2, wherethe total indentation work is shown as a function ofp;j2 for copper. It can be proved that even if thecontribution of the linear term in Equation 1 reaches40 per cent of the maximum load, the wtlP;j2 ratio

pI ,-~""- -- --- -- -- ---- ----------

h~

hm

(1)

Figure 1 Schematic picture of an indentation cycle: P, load; Pm'maximum load; Wd, work dissipated; Wc' work performed duringunloading; h, indentation depth.

4

3EE

~ 2$"

"8

oO 400 800 1000600200

p~2 (N3/2)

Figure 2 The total work, Wt' performed on copper depends linearly

on p~2 (where P ffi is the maximum load).

differs only by 7 per cent from the value calculatedfrom the quadratic function.

The quadratic term of the load-penetration depthfunction as a fraction of maximum load can be ex-pressed by the following

k x Pm = C3 X h~

According to the author's measurements for the differ-elit materials investigated, k varies between 0.6 and0.9. Similarly the cubic term in the total work functioncan be expressed as a fraction of the total work

k' x W - C3 h3t 3 x m

With the help of these functions the following relation-ship can be derived between Wt and p;j2

- ~ 1/2 1 3/2

Wt - k' k 3(C3)1/2Pm

Since C3is a constant depending only on the materialinvestigated, this relationship would be linear fora given material if (k/k')k1/2 were constant. Of coursethis condition is generally not strictly satisfied. Never-theless it can be shown (see Appendix) that in therange k ~ 0.6 this quantity varies only slightly, there-fore the Wt(p;j2) function is nearly linear. Conse-quently, using the formulae derived in the Appendix,the experimental results can be satisfactorily describedweil by the following relationships

$"---

SID

0.5

~1

.

0.00.0 0.5

(C3/ct)O5

Figure 3 The ratio of the elastic and total work versus the para-meters of the indentation curves: (6) glass, ("') Al, (T) Cu, (O) NaCl,(e) Si3N4, (O) plastic, (\7) A12O3-HAP, (O) Ce-TZP.

According to Equations 9-14 it is obvious that theworks and the parameters of the load-indentationdepth functions should also satisfy the following rela-tionship

(6)We

Wt - (C3/Cj)1/2(15)

Fig. 3 shows that the experimental data fit weil toEquation 15 for a broad variety of materials.

(7) 3.2. The connection between theconventional hardness number andthe parameters of the indentationcurves

The parameter C3which can be easily determined fromthe loading curve is generally used for the character-ization of the hardness of materials, however, it is notequal to the conventional hardness number [1, 4].

In the case of an ideally plastic material the load-indentation depth function is quadratic: P = c3h2[4, 13]. Since there is no elastic relaxation, thed = 7 hmequation between the diagonal of the Vickerspattern, d, and the indentation depth, hm,which is theconsequence of the geometry of the Vickers pyramid isexactly satisfied (Fig. 4a). Therefore the Vickers hard-ness of the ideally plastic materials obeys the followingequation

(8)

Pm PmBv = 1.8544 X -d2 = 1.8544 X - h2 = Cll X C3 (16)

49 X m

where Cll = 0.038.If the material is not ideally plastic then with in-

creasing contribution of the elastic deformation tothe total one, elastic deflection of the material underthe indenter increases (Fig. 4b). Consequently 7hmbe-comes step by step larger than d, and as a result of thisCllXC3will be less than Bv.

The authors have sought a relationship betweenBv and CllXC3in the form of

Bv = ~Cll C3 (17)

3111

Wt- m x p;j2 (9)

We = m* x p;j2 (10)

Wd - mx p;j2 (11)

where

m =1

3(C3)1/2(12)

m* -1

3(C3)1/2(13)

m = m - m* = [1 1 J3 (C3)1/2 - (cj)1/2(14)

PL1ld PlLlw,=WdWe=O

hw,= WeWd=O

h

~ d=7hm'~ ,

hm ~ElasticPlastic

Elastic

~(a) (b) (c)

Figure 4 Schematie pieture showing the behaviour of various ma-

terials during Viekers indentation: (a) ideally plastie, (b) elasto-plastie, and (e) ideally elastie.

It is obvious that ~must be one for an ideally plasticmaterial and with increasing elasticity it must in-crease; and in the limiting case of an ideally elasticmateriál it becomes infinite, because in this case thereis no residual deformation after unloading (Fig.4).The ~ = Wt/Wd ratio satisfies the conditions imposedon the limiting cases and according to the presentmeasurements the experimental results fit well fora broad variety of materials to the equation (Fig. 5)

WtC x-

Hv - CXlX 3 Wd(18)

3.3. The connection between theparameters of the indentation curvesand the Young's modul us

A similar relationship has been found between theparameters of the load-penetration depth curves andthe Young's modul us. According to the theoreticalinvestigation of Sneddon [11] the load-indentationdepth function for an ideally elastic material (Fig. 4c)can be described in both the loading and the unIoad-ing period by the same equation

P - c3h2 = ct h2 (19)

where

C3 = C* E cx3 o

2(1 y2)X2"xtan\f

- y(20)

where E is the Young's modul us, y is the Poisson'sratio, CXois a constant depending on the geometry ofthe indenter (cxo= 2 for Vickers indenter), y = (IT/2),lj!is the semi-angle 9f the indenter (74.05cfor a Vickersindenter) [11,13,14]. From this equation the Young'smodulus of ideally elastic materials can be given inthe form

E - CX2Ct (21)

with

2(1 - y2)y2

CX2 - tan \f CXo(22)

3112

h

20 ~ 0'4[2i0.3

0.2

01 L:Jl. 1

0.00.0 0.1 0.2 0.3

. .roc...~-; 10:J::

.°

o

'"~1

..

o 5 10

0.038 c3 w,1 Wd (GPa)

15

Figure 5 The eonventionally determined hardness number versusthe quantity caleulated on the basis of Equation 18: (6) glass,(A) Al, (T) Cu, (O) NaCl, (e) Si3N., (O) plastie, (O) Ce-TZP.

...

Substituting into these equations the parameters char-acterizing the geometry of the Vickers pyramid andtaking y as 1/3 one gets CX2= 0.63.Empiricalevidenceshows that Equation 21 is not valid for materialswhere the deformation is not ideally elastic. In thesecases CX2Ctbecomes higher than E. This can be ex-plained in the following way. In the course of theplastic deformation a plastic zone is developing underthe indenter (Fig. 4b) and due to this the elastic stres-ses and deformations are lower at the same E andPffi th an they would be in an ideally elastic material.Consequently, elastic relaxation is also smaller than inthe ideally elastic case. Assuming a relationship be-tweeD E and CX2ct in the form

E = WCX2ct (23)

the W factor must be one for ideally elastic materialsand it must decrease with increase of the plastic frac-tion of the deformation. Of course, in the ideallyplastic limit E must teDd to infinity. According to theexperimental results the ratio of the elastic and totalwork perforrned in the course of a loading-unloadingcycle can be used as W, i.e. Equation 23 can be writtenin the following form

We

E = CX2CtWt(24)

In the elastic limit this formula immediately yieldsEquation 21, predicted theoretically. In the ideallyplastic limit the validity of E -> 00 is not so obvious,because although ct -> 00, (We/Wt), as mentionedalready, tends to zero. However, rearranging Equa-tion 24 with the help of Equation 15 one can get

Wt

E - CX2C3We(25)

This formula implies E -> 00 for the ideally plasticlimiting case, since C3is a finite number and (Wt/We)tends to infinity. Fig. 6 shows that Youngs' modulimeasured by the four-point bending test agree rela-tively well with those calculated on the basis of Equa-tion 25. The deviation is especially significant for the

300

250

200coa..Q 150Lu

~11

100

50

O

o 150 25020050 100

0.63 c3 w,1 We (GPa)

Figure 6 Relationship between Young's modulus detennined byfour-point bending test and that calculated on the basis of Equation25: (ll) glass, (") Al, (T) Cu, (O) NaCl, (e) Si3N4, (D) plastic,(\7) AIP3-HAP, (O) Ce-TZP.

'"

~1

0.04 0.06 0.08

0.06 Wel Wd

0.10 0.12

Figure 7 The HylE ratio as a function of 0.06 (W,/Wd): (ll) glass,(") Al, (T) Cu, (O) NaCl, (e) Si3N4, (D) plastic, (O) Ce- TZP.

Alz03-HAP series which may be caused by takingv = 1/3 for the calculation of (Xz.

3.4. The isomechanical groupsAshby and Brown dassified materials iota iso-mechanical groups on the basis of the BvlE ratio [15].Oividing Equation 18 by Equation 25 one gets

Bv We- = (X3-E Wd

(26)

where (X3= 0.06. Fig.7 shows that the ratio of thehardness and the elastic modul us is proportional tothe ratio of the elastic and the dissipated work. Theexperimental results prove that using the work per-forrned in an indentation cyde, the Ashby type dassi-fication can be easily done because the indentationwork can be determined with great accuracy from thecontinuous indentation measurements. It is worth

mentioning that the C3parameter in Equations 18 and25 for the expressions of Bv and E, respectively, can becalculated with higher accuracy from the m, m* andmparameters of Equations 12-14 than directly fromthe load-indentation depth functions. The reason for

this is probably the uncertainty in the determinationof the starting point of the loading curve. Thereforeusing Equations 9-14 one has derived the followingformulae for the determination of Bv, E and the BvlEratio

4. ConclusionsA linear relationship was empirically found betweenthe indentation work determined in the loading andunloading period and P~z , where Pm wasthe maximum load of the indentation cyde. Relation-ships between the conventionally measured Vickershardness, Young's modulus and the parameters ofthe indentation curves were established. The equa-tions obtained empirically satisfy the conditionsimposed on the ideally elastic and plastic limitingcases.

It also has been proved that the ratio of the elasticand the dissipated work perforrned during an indenta-lion eyde can be used instead of the B viE ratio, on thebasis of which the materials can be dassified iotaisomechanical groups.

AppendixTo investigate the linearity of the Wt(P~z) functionthe quantity (klk')kI/Z can be expressed as a functionof k. Take the ratio of the two terms in Pm and Wt,respectively

(Al)

(A2)

From these equations one gets

k' = 2k3-k (A3)

Using Equation A3 one can obtain

~ k1/z = (3 - ~)kI/Z = f(k) (A4)

Fig. 8 showsf(k) as a function of k. It can be seen thatif k ~ 0.6 thenf(k) ~ 0.93.

It means that since for these present measurementsk ~ 0.6, f(k) is approximately constant which equalsto one and there is a linear connection between Wtand P~z. On the basis of Equations 2 and 4 a similarrelationship can be derived for We, i.e. it can be shownthat We(P~z) is also a linear function.

3113

0.12

0.10

0.08

Lu--; 0.06J::

0.04

0.02

0.000.00 0.02

Bv = (Xl9 mm (27)

E = (xz1

9 mm* (28)

Bv m*E = (X3 -=-

(29)m

k C3X h c3hm- -- --- -l-k czhm Cz

k' c3h:J3 2 x c3hm- - --1 - k' czhm/2 3 x Cz

1.0

0.93 ,- - - -- -- - - h -- -- -- -- -- -- - -- -- -- ~ - - -- -- - -_h -- -- h--

0.8

?~::;<I

s:2.

0.6

0.4

0.2

0.00.0 0.2 0.4 0.6

k

0.8 1.0

Figure 8 The f(k) function.

AcknowledgementsThe authors are grateful to M. Dimitrova-Lukács forproviding the AlzOrHAP and Ce-TZP ceramic sam-ples and Professor 1. Lendvai for very helpful dis-cussions and comm ents. This work was supported bythe Hungarian National Scientific Fund in contractnumber T-O17637, T-O17639 and T-O17474.

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Received 9 Augustand accepted 21 December 1995

i''1